AoPS Academy
+1 w
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
(i know 
well). i want to study so if you guys have some easy or normal problems please send me
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
with 
. Denote the incircle of 
as 
and let meet 
at 
respectively. Draw circle 
such that passes through 
and tangents to 
at 
. Denote the intersections of line 
and as 
and 
, the length of 
can be written in the simplest form of 
, compute 
.
has a right angle at 
. Let be the point on line 
such that 
and lies between and 
. Point 
is chosen so that 
and 
is the bisector of 
. Point 
is chosen so that 
and 
is the bisector of 
. Let 
be the midpoint of . Let be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
, and . The spheres with centers , and touch the plane at 
, and , respectively. Prove that 
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，and any rectangular strip of width 1 covers at most two points of S.
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
, 
, 
, 
, 
in some order.
be an acute triangle with orthocenter 
. Let 
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that bisects 
. Suppose that the line intersects the circumcircle of the triangle 
at and 
. Prove that 
.
, 
and 
be points on circle 
divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
and 
with sum 
. Show that![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
of the form 
with 
, consider the polynomial 
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
+1 w
satisfying: 
and 
Prove that 
be real numbers such that 
.
Prove that 
be real numbers such that 
.
Equality holds when 
or 
Equality holds when 
or 
Prove that
be real number . Find the minimum of 
Find the maximum of 
be real numbers. Prove that
T709. Let 
Prove that 
Let 
![$$ \frac{x+1}{\sqrt{3x^2+1}} + \frac{ x }{3x^2+1} \leq \frac{1}{\sqrt[3]2}+ \frac{1}{\sqrt[3]4}$$](http://latex.artofproblemsolving.com/8/1/4/814e258300d0409c135416044a26d95f0dd24dd1.png)
![\[
u=\frac{x}{\sqrt{3x^2+1}},\qquad v=\frac{1}{\sqrt{3x^2+1}},
\]](http://latex.artofproblemsolving.com/5/4/5/5453da04887ddeaf134c0eea90868a479048090c.png)
then![\[
3u^2 + v^2 = 1.
\]](http://latex.artofproblemsolving.com/4/0/a/40a37e6a51dabcf5ab61903166e3fd09284d8f31.png)
![\[
\begin{aligned}
\text{RHS} - \text{LHS} &= \frac{3}{2} - (u+v) - 2u^2 \\
&=\frac{1}{2} + (1-2u^2) - (u+v) \\
&=\frac{1}{2} + (u^2+v^2) - (u+v) \\
&=\left(u-\frac{1}{2}\right)^2+\left(v-\frac{1}{2}\right)^2 \ge 0.
\end{aligned}
\]](http://latex.artofproblemsolving.com/7/5/f/75f224a32d8d4187689af7077dca0e19c3e90523.png)
![$u = \frac{x}{\sqrt{3x^2+1}}, v = \frac{1}{\sqrt{3x^2+1}}, t = \sqrt[3]{2}$](http://latex.artofproblemsolving.com/e/4/c/e4cecb02ce3fb081e481ffc2cb2209ef73f1f48e.png)
,
